Principled Analyses and Design of First-Order Methods with Inexact Proximal Operators

Proximal operations are among the most common primitives appearing in both practical and theoretical (or high-level) optimization methods. This basic operation typically consists in solving an intermediary (hopefully simpler) optimization problem. In this work, we survey notions of inaccuracies that can be used when solving those intermediary optimization problems. Then, we show that worst-case guarantees for algorithms relying on such inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach introduced by Drori and Teboulle (Mathematical Programming, 2014) and on convex interpolation results, and allows producing non-improvable worst-case analyzes. In other words, for a given algorithm, the methodology generates both worst-case certificates (i.e., proofs) and problem instances on which those bounds are achieved. Relying on this methodology, we provide three new methods with conceptually simple proofs: (i) an optimized relatively inexact proximal point method, (ii) an extension of the hybrid proximal extragradient method of Monteiro and Svaiter (SIAM Journal on Optimization, 2013), and (iii) an inexact accelerated forward-backward splitting supporting backtracking line-search, and both (ii) and (iii) supporting possibly strongly convex objectives. Finally, we use the methodology for studying a recent inexact variant of the Douglas-Rachford splitting due to Eckstein and Yao (Mathematical Programming, 2018). We showcase and compare the different variants of the accelerated inexact forward-backward method on a factorization and a total variation problem.Comment: Minor modifications including acknowledgments and references. Code available at https://github.com/mathbarre/InexactProximalOperator

A note on approximate accelerated forward-backward methods with absolute and relative errors, and possibly strongly convex objectives

In this short note, we provide a simple version of an accelerated forward-backward method (a.k.a. Nesterov's accelerated proximal gradient method) possibly relying on approximate proximal operators and allowing to exploit strong convexity of the objective function. The method supports both relative and absolute errors, and its behavior is illustrated on a set of standard numerical experiments. Using the same developments, we further provide a version of the accelerated proximal hybrid extragradient method of Monteiro and Svaiter (2013) possibly exploiting strong convexity of the objective function.Comment: Minor modifications in notations and acknowledgments. These methods were originally presented in arXiv:2006.06041v2. Code available at https://github.com/mathbarre/StronglyConvexForwardBackwar

Topographically induced internal solitary waves in a pycnocline: Ultrasonic probes and stereo-correlation measurements

Internal solitary waves (ISWs) are large amplitude stable waves propagating in regions of high density gradients such as the ocean pycnocline. Their dynamics has often been investigated in two-dimensional approaches, however, their three-dimensional evolution is still poorly known. Experiments have been conducted in the large stratified water tank of CNRM-GAME to study the generation of ISWs in two academic configurations inspired by oceanic regimes. First, ultrasonic probes are used to measure the interfacial displacement in the two configurations. In the primary generation case for which the two layers are of constant density, the generation of ISWs is investigated in two series of experiments with varying amplitude and forcing frequency. In the secondary generation case for which the lower layer is stratified, the generation of ISWs from the impact of an internal wave beam on the pycnocline and their subsequent dynamics is studied. The dynamics of ISWs in these two regimes accords well with analytical approaches and numerical simulations performed in analogous configurations. Then, recent developments of a stereo correlation technique are used to describe the three-dimensional structure of propagating ISWs. In the primary generation configuration, small transverse effects are observed in the course of the ISW propagation. In the secondary generation configuration, larger transverse structures are observed in the interfacial waves dynamics. The interaction between interfacial troughs and internal waves propagating in the lower stratified layer are a possible cause for the generation of these structures. The magnitude of these transverse structures is quantified with a nondimensional parameter in the two configurations. They are twice as large in the secondary generation case as in the primary generation case

Particle Effects on the ISGRI Instrument On-Board the INTEGRAL Satellite

The INTEGRAL satellite was launched on October 17, 2002. All on-board instruments are operating successfully. In this paper, we focus on radiation effects on the Cadmium Telluride camera ISGRI. The spectral response of the camera is affected by cosmic particles depositing huge amount of energy, greater than the high threshold of the electronics. Our study raises the contribution of cosmic ray protons. Solutions are proposed to limit the degradation of spectral response of large pixel gamma cameras operating in space

Petrography and Geochemistry of the Intrusive Rocks at the Diorite-Hosted Regnault Au Mineralization

Archean greenstone belts are renowned for their Au endowment. Gold can be associated with orogenic Au style of mineralization and with diorite-hosted Au-Cu disseminated sulfides interpreted as a porphyry style of mineralization. The Regnault Au project, located in the Frotet–Evans greenstone belt (Superior craton), is a structurally-controlled and diorite-hosted mineralization with an unclear metallogenic model. The aim of this study is to evaluate the fertility of the Regnault granodiorite-diorite-gabbro intrusive. Using whole-rock chemistry and petrological observations, it is concluded that the intrusive suite derives from a differentiated and water-bearing magma extracted from the metasomatized mantle. Amphibole chemistry indicates that the magma was moderately oxidized and that it emplaced at a shallow depth. It is concluded that the Regnault intrusive suite displays several characteristics favorable to the exsolution of magmatic fluids in the upper crust and that the Regnault Au mineralization potentially corresponds to, at least in part, a magmatic-hydrothermal system

The assessment of the use of eco-friendly nets to ensure sustainable cabbage seedling production in Africa

High seed cost accompanied by poor germination and seedling performance renders cabbage nursery and field production enterprises unsustainable to many small-scale growers in tropical and sub-tropical countries. In most nurseries, adverse ecological conditions and pest damage are among the major factors responsible for poor seedling performance. The objective of this study was to test the potential use of eco-friendly net (EFN) covers as a low cost technology for sustainable cabbage seedling production. The study was a two-season experiment conducted using a randomized complete block design with five replications and two treatments. Treatments were: the standard open field transplant production (control) and transplant production under 0.4 mm mesh polyethylene net covering. EFN covering increased both temperature and relative humidity, enhanced seedling growth and reduced insect pest damage. Seed germination and seedling emergence were under the net covering earlier. Higher seed germination and seedling survival were recorded under the EFN treatment, indicating a potential for reducing the seed requirement per unit area of cabbage production. Seedlings grown under the nets had higher stomatal conductance and leaf chlorophyll content; furthermore, they grew taller, with more leaves within a shorter period of time compared to the control seedlings. The use of EFN in cabbage nurseries offers a sustainable technology for enhancing seedling performance by reducing pest infestation, thereby lowering production cost and improving the grower's income. (Résumé d'auteur